You would just need the issuers of the local paper currency to be able to show and prove backing for the paper currency in a publicly auditable fashion. This means the issuer(s) would have to put up the bitcoin first before issuing the local paper currency. The purchasing power of the local currency would fluctuate with the value of bitcoin unless you adjusted the amount of bitcoin backing regularly so that the local paper currency is always traded 1 to 1 with the US dollar.